To run a bash script, first create a file with your commands. For example, a file named hello.sh with the content '#!/bin/bash' and 'echo "Hello, world!"'. Next, make the script executable using 'chmod +x hello.sh'. This allows you to run it directly with './hello.sh'. When you run it, the script executes the commands inside and prints 'Hello, world!' to the terminal. The script ends and the terminal prompt returns. Alternatively, you can run the script without making it executable by using 'bash hello.sh'. The first line '#!/bin/bash' tells the system to use bash to run the script. Without execute permission, running './hello.sh' will give a permission denied error.